Ordinary varieties and the comparison between multiplier ideals and test ideals. II.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1256.13003
DOI: 10.1090/S0002-9939-2011-11240-1
author: Mircea Mustaţă
published in: Proceedings of the American Mathematical Society
publication date: 2012-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.2915
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 14F18
Mathematics Subject Classification ID: 14F30
zbMATH DE Number: 6024088
zbMATH Keywords: multliplier ideal
zbMATH Keywords: test ideal
zbMATH Keywords: ordinary variety
MaRDI profile type: Publication
arXiv ID: 1012.2915
cites work: $F$-thresholds of hypersurfaces
cites work: Discreteness and rationality of \(F\)-thresholds
cites work: \(p\)-adic étale cohomology
cites work: Initial ideals, Veronese subrings, and rates of algebras
cites work: F-Purity and Rational Singularity
cites work: A generalization of tight closure and multiplier ideals
cites work: Q5694889
cites work: Test Ideals Vs. Multiplier Ideals
OpenAlex ID: W2052664492
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title: Ordinary varieties and the comparison between multiplier ideals and test ideals. II.
review text: In this paper, the author relates the following two conjectures. For simplicity of notation in this review, we consider our varieties with defining equations in \(\mathbb{Z}\). The paper works in greater generality. In particular, the first conjecture is as follows.NEWLINENEWLINEConjecture 1.1. For every ideal \(a\) on a smooth variety \(X\) of characteristic \(0\), there exists a Zariski-dense set of prime integers such that if \(X_p\) (respectively \(a_p\)) is the reduction mod \(p\) of \(X\) (respectively \(a\)) then \(\mathcal{J}(X, a^t)_p = \tau(X_p, a_p^t)\) for all \(t \geq 0\). Here \(\mathcal{J}(X, a^t)\) is the multiplier ideal and \(\tau(X_p, a_p^t)\) is the test ideal.NEWLINENEWLINEThe second conjecture states:NEWLINENEWLINEConjecture 2. For every smooth projective \(d\)-dimensional variety \(X\) of characteristic \(0\), there is a dense set of primes such that the Frobenius induces isomorphisms \(H^{d}(X_p, \mathcal O_{X_p}) \to H^{d}(X_p, \mathcal O_{X_p})\).NEWLINENEWLINEThis bijectivity of Frobenius is weaker than the assertion that \(X_p\) is ordinary in the sense of \textit{S. Bloch} and \textit{K. Kato} [Publ. Math., Inst. Hautes Étud. Sci. 63, 107--152 (1986; Zbl 0613.14017)].NEWLINENEWLINEIn [Nagoya Math. J. 204, 125--157 (2011; Zbl 1239.14011)], the author and \textit{V. Srinivas} related two these conjectures. They showed that conjecture 1.2 implies conjecture 1.1. In this short paper, the author proves that conjecture 1.1 implies conjecture 1.2 showing that the two conjectures are equivalent.
reviewed by: Karl Schwede
description: scientific article; zbMATH DE number 6024088